Dispersion Modelling The advectiondiffusion equation

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Dispersion ModellingThe advection-diffusion
equation

• Conservation of mass for a simple box

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Mass balances

• input output accumulation
• Total mass
• Contaminant
• Advection, diffusion, dispersion mechanisms

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Diffusive and dispersive fluxes
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Diffusive and dispersive fluxes

• Diffusivity, DAB, is determined by temperature,
  pressure, and molecular properties of A and B
  molecules.
• Dispersion coefficients Di (or Ki), ix, y, z
• are determined by flow conditions (velocity,
  intensity, and scale of turbulence) and are
  defined consistently with diffusive fluxes,

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Diffusive and dispersive fluxes

• Advective, diffusive and dispersive fluxes are
  additive.
• In atmospheric transport dispersive flux
  dominates diffusive flux.
• The relation between advective and dispersive
  flux is case dependent, one or the other may
  dominate in different directions.

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• Considering both advective and dispersive fluxes,
  we can get (for the contaminant)
• Note the positioning of the terms to avoid the
  appearance of negative signs.
• Since the diffusive and dispersive fluxes are
  defined similarly, if we let Dx represent the sum
  of diffusivity and dispersion coefficient in the
  x direction the equation will represent all three
  processes. As already noted, dispersion
  typically dominates anyway in many environmental
  situations.

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• Other process may also be considered
• Generation within box (emissions)
• Chemical reaction (production or decay)
• Surface uptake (plants?)
• Deposition (wet or dry)

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• ci concentration of pollutant i,
• a function of space (x,y,z) and time (t)
• u,v,w horizontal and vertical wind speed
  components
• KX, KY horizontal turbulent diffusion
  coefficients
• KV vertical turbulent exchange coefficients
• Ri net rate of production of pollutant i by
  chemical reactions
• Si emission rate of pollutant i
• Di net rate of change of pollutant i due to
  surface uptake processes
• Wi net rate of change of pollutant i due to
  wet deposition

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Dispersion from a point source
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Figure 4-1 Wark Warner

• Gaussian or normal distribution function

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GAUSSIAN (NORMAL) DISTRIBUTION
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DOUBLE GAUSSIAN DISTRIBUTION
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• Can view the plume two ways
• as a person standing on the ground
• - Eulerian space
• - contaminant will move as a result of advection
  and dispersion
• 2) as a person moving with the air mass
• - Lagrangian space
• - contaminant will move as a result of
  dispersion

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3 DIMENSIONAL DISPERSION MODELLagrangian
approach to the point source problem

• Similar to heat conduction equation in 3-d
• Solution for instantaneous release of X g of
  pollutant at t 0 and x y z 0

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Lagrangian solutions to the diffusion equation
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Eulerian viewpoint for an elevated point source

• Consider advection in only one direction
• u wind speed in x direction
• v w 0
• Consider a release of Q g/s over a short time
  period ?t at time zero, thus, X Q ?t
• Let the point of release be at (0, 0, H)
• At time t, the center of the dispersing puff will
  be at (ut, 0, H)

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• If we wish to consider a continuous emission
  source and evaluate the lateral spreading in the
  y and z directions only, the equation becomes
•  
•  
•  
• where Q mass emission rate, mass/time
• u advection velocity, length/time
• See next slide from de Nevers for derivation

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Dispersion coefficients

• Define dispersion coefficients sy and sz in terms
  of Ky and Kz, u and x

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2-D STEADY DISPERSION MODEL

• Solution for windspeed of u m/s and continuous
  release of Q g/s of pollutant at x y 0
  (stack location) and z H (the effective
  stack height)
• H h ??h
• h physical stack height, ?
• ?h plume rise due to buoyancy

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DISPERSION COEFFICIENTS

• Ky and Kz approximately proportional to wind
  speed
• Ky/u and Kz/u approximately constant
• ?y and ?z should vary approximately with
  x(1/2)
• Field observations show more complex variation
  (Figures 4.6 and 4.7, Wark, Warner Davis)
• Wind speed and solar flux combine to give
  stability classes A - F (Table 6.1 de Nevers)

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• Horizontal dispersion coefficient

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• Vertical dispersion coefficient

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Algebraic expressions for dispersion coefficients

• The graphical representations of ?y and ?z can
  be approximated by algebraic expressions using
  parameters in Tables 4-1 through 4-3 of Wark,
  Warner, and Davis.

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Algebraic representation of dispersion
coefficients Rural environment
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Algebraic representation of dispersion
coefficients Urban environment
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Algebraic representation of dispersion
coefficients Urban environment
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ATMOSPHERIC TURBULENCE AND SAMPLING TIME

• The time scale for atmospheric turbulence can be
  quite long, of the order of many minutes.
• Field observations of dispersion coefficients are
  specific to the sampling (averaging) time used
  (typically 10 minutes)
• Estimates of corrections for other sampling
  periods can be made

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